If the order of the multiplicands counts for different ways, such as 5 * 625 * 390625 counting as a different way to 390625 * 625 * 5, then there's a simple way of calculating the ways: If the factors are represented as their respective power of 5 (such as 1,4,8 or 8,4,1 in the above examples) then the total powers of 5 has to be 13. These numbers can be represented by groups of one 1, four 1's and 8 1's. All together, these 13 1's, together with two separators (to separate which go into which power) make 15 items of which two are separators. There are C(15,2) ways of doing this, or 105. Similarly, for 5^12, there are C(14,2) = 91 ways.

However if order of factors is not considered to create a new way, then I don't see an alternative to enumerating them. For 13 and 12, the below tables enumerate, first the powers (exponents) of 5 themselves, for simplicity, and then the numbers themselves. The latter tables show also the number of ways of rearranging each, with the total of those matching the combination method outlined above.